| L(s) = 1 | + 5-s − 3·7-s + 3·11-s − 4·13-s − 5·17-s − 19-s − 4·25-s − 2·29-s + 8·31-s − 3·35-s − 10·37-s − 6·41-s − 7·43-s + 9·47-s + 2·49-s + 8·53-s + 3·55-s − 14·59-s − 5·61-s − 4·65-s + 6·71-s − 15·73-s − 9·77-s − 4·79-s − 4·83-s − 5·85-s + 12·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.904·11-s − 1.10·13-s − 1.21·17-s − 0.229·19-s − 4/5·25-s − 0.371·29-s + 1.43·31-s − 0.507·35-s − 1.64·37-s − 0.937·41-s − 1.06·43-s + 1.31·47-s + 2/7·49-s + 1.09·53-s + 0.404·55-s − 1.82·59-s − 0.640·61-s − 0.496·65-s + 0.712·71-s − 1.75·73-s − 1.02·77-s − 0.450·79-s − 0.439·83-s − 0.542·85-s + 1.25·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.227970694274940482118975475364, −8.630197680642780580590235451603, −7.35801470679317747358663157370, −6.65084427660654519125997905895, −6.07596938444397512729377964410, −4.93646030803797952754928020473, −3.98454062268883454433648080079, −2.92269314971800745710061052414, −1.85925474650503174098776451275, 0,
1.85925474650503174098776451275, 2.92269314971800745710061052414, 3.98454062268883454433648080079, 4.93646030803797952754928020473, 6.07596938444397512729377964410, 6.65084427660654519125997905895, 7.35801470679317747358663157370, 8.630197680642780580590235451603, 9.227970694274940482118975475364
